Salut!
Pentru a rezolva exercitiul, folosim reguli de calcul cu puteri! In caz ca nu le stii, acestea sunt cele mai importante care le folosim in acest exercitiu:
[tex]\text{Regula 1: $a^m\cdot a^n=a^{m+n}$}\\\text{Regula 2: $a^m\div a^n = a^{m-n}$}\\\text{Regula 3: $(a^m)^n=a^{mn}$}[/tex]
a) [tex]2^2 \cdot 2^3 \cdot 2^4 \cdot 5^2 \cdot 5^4 = 2^9 \cdot 5^6 = 2^3 \cdot 2^6 \cdot 5^6 = 2^3 \cdot 10^6 = 8 \cdot 10^6[/tex]
b) [tex]2^4 \cdot 2^4 \cdot 125 \cdot 25^3 = (2^4)^2 \cdot 5^4 \cdot (5^2)^3 = 2^8 \cdot 5^4 \cdot 5^6 =1 6 \cdot 5^4 \cdot 5^6 = 16 \cdot 5^{10}[/tex]
c) [tex]2^1 \cdot 2^2 \cdot 2^3 \cdot 3^2 \cdot 5^6 = 2^6 \cdot 3^2 \cdot 5^6 = 64 \cdot 9 \cdot 5^6 = 576 \cdot 5^6[/tex]
d) [tex]\begin{aligned}2^1 \cdot 2^2 \cdot 2^3 \cdot 2^4 \cdot 5^1 \cdot 5^2 \cdot 5^3 \cdot 5^4 \div 10^{10} = 2^{10} \cdot 5^{10} \div 10^{10} = \frac{2^{10}\cdot5^{10}}{10^{10}}& = \frac{(2\cdot5)^{10}}{10^{10}} =\\\frac{10^{10}}{10^{10}}& = 1\\\end{aligned}[/tex]
e) [tex]\begin{aligned}64 \cdot 16 \cdot 32 \cdot 125 \cdot 125 \cdot 625 \cdot 625 \cdot 5 \div 10^{13} &= 2^6 \cdot 2^4 \cdot 2^5 \cdot 5^3 \cdot 5^3 \cdot 5^4 \cdot 5^4 \cdot 5 \div 10^{13} \\& = 2^{15}\cdot 5^{17}\div10^{13}\\& = \frac{2^{15}\cdot5^{17}}{10^{13}}\\& = \frac{2^{15}\cdot5^{17}}{5^{13}\cdot2^{13}}\\& = 2^2 \cdot 5^4 = 4 \cdot 625 = 2500\end{aligned}[/tex]
f) [tex]\begin{aligned}8 \cdot 16 \cdot 128 \div 4 \div 32 \div 64 \cdot 3125 \div 625 & = 2^3 \cdot 2^4 \cdot 2^7 \div 2^2 \div 2^5 \div 2^6 \cdot 5^5 \div 5^4\\& = 2^{14}\div2^2\div 2^5 \div 2^6 \cdot 5\\& = 2 \cdot 5 = 10\end{aligned}[/tex]
-Luke48
